The conjecture that general-type varieties have no nowhere-vanishing holomorphic one-forms

Let $X$ be a smooth projective variety of general type. A nowhere vanishing holomorphic one-form on $X$ is a holomorphic section of the cotangent bundle $4\Omega_X^14$ with no zero. The general-type one-form conjecture. $X$ does not admit a nowhere vanishing holomorphic one form. This is known for canonically polarized varieties and for threefolds, but remains open for smooth projective varieties of general type in general.